Measurable Riemannian geometry on the Sierpinski gasket: the Kusuoka measure and the Gaussian heat kernel estimate

We study the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as a prototype of "measurable Riemannian geometry". The shortest pathmetric on the harmonic Sierpinski gasket is shown to be the geodesic distance associated with the "measurable Riemannian structure". The Kusuoka measure is shown to have the volume doubling property with respect to the Euclidean distance and also to the geodesic distance. Li-Yau type Gaussian off-diagonal heat kernel estimate is established for the heat kernel associated with the Kusuoka measure.